Efficient Algorithms And Applications Of Low-rank Tensor Completion

With the rapid development and wide application of information technology,the huge value of massive data has attracted the attention of many researchers.As an important data analysis tool,low-rank tensor completion has become a hot topic in the field of computer vision,artificial intelligence and optimization.It mainly fills in data that is partially lost,contains noise or outliers,and has important applications in im-age/video repair,signal reconstruction,and data mining.However,the existing algorithm has the problems such as slow speed and small scale of problems that can be handled,making it unable to exert its advantages.In addition,in many applications,the prior information of the actual data is particularly important for the establishment of the model.This paper will focus on the models,algorithms and applications in the low rank tensor completion.The main contents are as follows:Chapter One introduces the research background and significance of low-rank matrix completion and low-rank tensor completion,as well as the development and relationship between low-rank matrix completion and low-rank tensor completion.And mathematical concepts that are to be adopted in the following chapters are given.Finally,the arrangement of the main work in this paper is explained.Chapter Two designs a proximity algorithm with adaptive penalty for nuclear norm minimization to better solve the subproblems with alternate direction method in the low-rank matrix completion based on the nuclear norm minimization method.Firstly,a unified model is adopted to represent three kinds of nuclear norm minimization problems.Then,an improved alternate direction method is designed to solve the model,that is,an adaptive penalty term is added to one of the sub-problems in case there is no closed solution.The penalty parameter is set in accordance with the adaptive update strategy.In addition,the subdifferential and proximity operator are used to construct the equivalent fixed-point equations applied to prove the convergence of the algorithm.Finally,the results of simulation and real-value experiments show that the method proposed in this chapter is feasible and effective compared with the other methods for nuclear norm minimization.Chapter Three discusses tensor completion based on low rank matrix decomposition.Most of the exist-ing low-rank tensor completion models make insufficient use of the observed data.As a result,the established models fail to take into account the piecewise smoothness of the data,or are excessively sparse,which leads to the subtle features of the recovered data be ignored.Therefore,by using the (?) norm of low-rank matrix de-composition and frame transformation,this chapter establishes the sparse low-rank tensor completion model,further designs the solution model of block successive upper bound minimization algorithm,and analyzes the convergence of the algorithm.Finally,The results of numerical experiments show that the method proposed in this chapter has a good effect on high-dimensional image reconstruction.